Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a doubly stochastic matrix $B'$ with entries $\beta_{ii}$, does there exist a unitary matrix with entries $\alpha_{ii}$ such that $|\alpha|^2_{ii} = \beta_{ii}$?

"In this paper, it is shown that for n⩾3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2]"

Interesting topic. Today we do not have a clear picture about the relationship between being uni-stochastic (or ortho-stochastic, if you restrict your attention to orthogonal matrices) and doubly-stochastic.